Triangle calculator SSA

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Triangle has two solutions with side c=0.92220152767 and with side c=0.39877157529

#1 Acute scalene triangle.

Sides: a = 1.06   b = 0.87   c = 0.92220152767

Area: T = 0.38224356385
Perimeter: p = 2.85220152767
Semiperimeter: s = 1.42660076384

Angle ∠ A = α = 72.46330032977° = 72°27'47″ = 1.26547179934 rad
Angle ∠ B = β = 51.5° = 51°30' = 0.89988445648 rad
Angle ∠ C = γ = 56.03769967023° = 56°2'13″ = 0.97880300954 rad

Height: ha = 0.72215766763
Height: hb = 0.87991623872
Height: hc = 0.83295646463

Median: ma = 0.72329149917
Median: mb = 0.89331019456
Median: mc = 0.85330662092

Inradius: r = 0.26881862482
Circumradius: R = 0.5565833716

Vertex coordinates: A[0.92220152767; 0] B[0; 0] C[0.66598655148; 0.83295646463]
Centroid: CG[0.52772935972; 0.27765215488]
Coordinates of the circumscribed circle: U[0.46110076384; 0.3110520655]
Coordinates of the inscribed circle: I[0.55660076384; 0.26881862482]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.5376996702° = 107°32'13″ = 1.26547179934 rad
∠ B' = β' = 128.5° = 128°30' = 0.89988445648 rad
∠ C' = γ' = 123.9633003298° = 123°57'47″ = 0.97880300954 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 1.06 ; ; b = 0.87 ; ; beta = 51° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 0.87**2 = 1.06**2 + c**2 -2 * 1.06 * c * cos (51° 30') ; ; ; ; c**2 -1.32c +0.367 =0 ; ; p=1; q=-1.32; r=0.367 ; ; D = q**2 - 4pr = 1.32**2 - 4 * 1 * 0.367 = 0.274889990678 ; ; D>0 ; ; ; ;
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1.32 ± sqrt{ 0.27 } }{ 2 } ; ; c_{1,2} = 0.65986551 ± 0.26214976191 ; ; c_{1} = 0.922015276746 ; ; c_{2} = 0.397715752926 ; ; ; ; text{ Factored form: } ; ; (c -0.922015276746) (c -0.397715752926) = 0 ; ; ; ; c>0 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 1.06 ; ; b = 0.87 ; ; c = 0.92 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 1.06+0.87+0.92 = 2.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.85 }{ 2 } = 1.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.43 * (1.43-1.06)(1.43-0.87)(1.43-0.92) } ; ; T = sqrt{ 0.15 } = 0.38 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.38 }{ 1.06 } = 0.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.38 }{ 0.87 } = 0.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.38 }{ 0.92 } = 0.83 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 0.87**2+0.92**2-1.06**2 }{ 2 * 0.87 * 0.92 } ) = 72° 27'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.06**2+0.92**2-0.87**2 }{ 2 * 1.06 * 0.92 } ) = 51° 30' ; ;
 gamma = 180° - alpha - beta = 180° - 72° 27'47" - 51° 30' = 56° 2'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.38 }{ 1.43 } = 0.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.06 }{ 2 * sin 72° 27'47" } = 0.56 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.87**2+2 * 0.92**2 - 1.06**2 } }{ 2 } = 0.723 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.92**2+2 * 1.06**2 - 0.87**2 } }{ 2 } = 0.893 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.87**2+2 * 1.06**2 - 0.92**2 } }{ 2 } = 0.853 ; ;



#2 Obtuse scalene triangle.

Sides: a = 1.06   b = 0.87   c = 0.39877157529

Area: T = 0.16549654639
Perimeter: p = 2.32877157529
Semiperimeter: s = 1.16438578765

Angle ∠ A = α = 107.5376996702° = 107°32'13″ = 1.87768746602 rad
Angle ∠ B = β = 51.5° = 51°30' = 0.89988445648 rad
Angle ∠ C = γ = 20.96330032977° = 20°57'47″ = 0.36658734287 rad

Height: ha = 0.31112555923
Height: hb = 0.37992309516
Height: hc = 0.83295646463

Median: ma = 0.42202843205
Median: mb = 0.67220594543
Median: mc = 0.949905508

Inradius: r = 0.14217402135
Circumradius: R = 0.5565833716

Vertex coordinates: A[0.39877157529; 0] B[0; 0] C[0.66598655148; 0.83295646463]
Centroid: CG[0.35325270893; 0.27765215488]
Coordinates of the circumscribed circle: U[0.19988578765; 0.51990439912]
Coordinates of the inscribed circle: I[0.29438578765; 0.14217402135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.46330032977° = 72°27'47″ = 1.87768746602 rad
∠ B' = β' = 128.5° = 128°30' = 0.89988445648 rad
∠ C' = γ' = 159.0376996702° = 159°2'13″ = 0.36658734287 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 1.06 ; ; b = 0.87 ; ; beta = 51° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 0.87**2 = 1.06**2 + c**2 -2 * 1.06 * c * cos (51° 30') ; ; ; ; c**2 -1.32c +0.367 =0 ; ; p=1; q=-1.32; r=0.367 ; ; D = q**2 - 4pr = 1.32**2 - 4 * 1 * 0.367 = 0.274889990678 ; ; D>0 ; ; ; ; : Nr. 1
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1.32 ± sqrt{ 0.27 } }{ 2 } ; ; c_{1,2} = 0.65986551 ± 0.26214976191 ; ; c_{1} = 0.922015276746 ; ; c_{2} = 0.397715752926 ; ; ; ; text{ Factored form: } ; ; (c -0.922015276746) (c -0.397715752926) = 0 ; ; ; ; c>0 ; ; : Nr. 1
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 1.06 ; ; b = 0.87 ; ; c = 0.4 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 1.06+0.87+0.4 = 2.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.33 }{ 2 } = 1.16 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.16 * (1.16-1.06)(1.16-0.87)(1.16-0.4) } ; ; T = sqrt{ 0.03 } = 0.16 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.16 }{ 1.06 } = 0.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.16 }{ 0.87 } = 0.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.16 }{ 0.4 } = 0.83 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 0.87**2+0.4**2-1.06**2 }{ 2 * 0.87 * 0.4 } ) = 107° 32'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.06**2+0.4**2-0.87**2 }{ 2 * 1.06 * 0.4 } ) = 51° 30' ; ;
 gamma = 180° - alpha - beta = 180° - 107° 32'13" - 51° 30' = 20° 57'47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.16 }{ 1.16 } = 0.14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.06 }{ 2 * sin 107° 32'13" } = 0.56 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.87**2+2 * 0.4**2 - 1.06**2 } }{ 2 } = 0.42 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.4**2+2 * 1.06**2 - 0.87**2 } }{ 2 } = 0.672 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.87**2+2 * 1.06**2 - 0.4**2 } }{ 2 } = 0.949 ; ;
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